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Whittling down RADs. Ken Locey. These are the only RADs possible for the community you think you have. Q. How many possible RAD’s for a community of 45 individuals?. Q. How many possible RADs for a community of 45 individuals?. A. 89, 134.
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Whittling down RADs • Ken Locey These are the only RADs possible for the community you think you have
Q. How many possible RAD’s for a community of 45 individuals?
Q. How many possible RADs for a community of 45 individuals? A. 89, 134
Q. How many possible ways to add integers to reach a sum of 45? A. 89, 134
Integer partitioning: number of unordered ways to add integers to sum to another integer • 4: • 1,1,1,1 • 4 • 2,2 • 1,3 • 3: • 1,1,1 • 3 • 2,1 • 5: • 1,1,1,1,1 • 5 • 2,2,1 • 1,3,1 • 2,3 • 4,1 • 2,1,1,1
Partitioning Community of size N Community of size N Integer of size N Can have 1 to N summands The summands can take values 1 to N Values of integers must sum to N • Can have 1 to N species • The species can take values of 1 to N • Abundances of species must sum to N
Community State Variables = Partition State Variables A community An integer partition Integer value Partition length Largest integer in partition 1’s in the partition • Community size = • Community richness = • Most abundant species = • Singleton species =
N = 30 # partitions (i.e. RADs) = 5604
N = 30, S = 4 # partitions (i.e. RADs) = 297
N = 30, S = 4, max = 10 # partitions (i.e. RADs) = 23 from 5604 99.6% decrease
N = 45, S = 5, max <= 25, min = 1 # RADs = 458
N = 45, S = 5, max <= 25, min = 1 # RADs = 458 from 89,134 99.5% decrease
N = 45, S = 5, max = 25, min = 1 # RADs = 30 from 89,134 99.97% decrease
Having community size… • A community of 100 individuals will have 190,569,292 possible RAD’s
Having community size… • A community of 100 individuals will have 190,569,292 possible RAD’s • If we know: • S= 8, max <= 50, at least 1 singleton: • 487,182 possibilities • 99.7% decrease • Imagine if we knew a little more?
Point • # RADs for a community can be large but the number can be found • Knowing community size and 3 more pieces of information can whittle down the possibilities by more than 99.5%. • If you want to predict the distribution of abundance, you do not need to know much • What you know should be accurate
Challenges • Computer algorithms fast enough to yield answers in a reasonable time • Storage is not the problem • Identifying which features whittle-down the possibilities the most. • Most informative state variables • Graphical examination
Within these RADs are the only RADs possible …for the community you think you have